Comparizer Ideals of Rings

Abstract

A right ideal I of a ring R is called a comparizer right ideal if for all right ideals A, B of R, either A ⊆ B or BI ⊆ A. For every ring R there exists the largest comparizer ideal C ₁(R) of R, and higher comparizer ideals C _α(R) can be defined inductively. In this paper, comparizer right ideals and relationships between the iterated comparizer ideal and some classical radicals of R are studied. Obtained results generalize some properties of right chain rings, right distributive rings, right Bezout rings and rings with comparability.

Key Words:

• Comparizer ideal
• Prime radical
• Generalized nil radical
• Strongly prime radical
• Right chain ring
• Right distributive ring
• Right Bezout ring

Mathematics Subject Classification:

• Primary 16D25, 16D70
• Secondary 16N40

Acknowledgments

The first author was supported by KBN Grant 5 P03A 04120. Most of this work was done when the first author was visiting University of Duisburg. Here he wishes to thank for the hospitality given by the members of the Department of Mathematics.

Notes

^#Communicated by E. Puczyłowski.